High-speed data communication may be provided through the use of multi-carrier transmission systems. However, multi-carrier transmission systems generally have a high peak-to-average power ratio (PAR), which is undesirable. PAR increases in proportion to the number of subcarriers and the order of constellations. However, peak power is limited by frequency regulation agencies, such as the Federal Communication Commission in the United States. Moreover, a high PAR corresponds to a relatively wide dynamic range in transceivers. Due to realization limitations, multi-carrier signals are typically amplified with a large back off in radio frequency power amplifiers. This reduces both the power efficiency and the average power emission, resulting in a decrease in the transmission range.
The peak power of a multi-carrier transmission system is defined to be the power of a sine wave with an amplitude equal to the maximum envelope. Hence, PAR is defined to be the ratio of the peak power to the average power (see R. van Nee and R. Prasad, OFDM Wireless Multimedia Communications, Artech House, 2000).
However, because the peak power has large variations according to the statistical characteristics of transmitted information symbols, a statistical metric is generally used to determine PAR (see X. Li and L. J. Cimini, “Effects of clipping and filtering on the performance of OFDM,” IEEE Communications Letters, vol. 2, no. 5, pp. 131–133, May 1998). Thus, PAR can be determined based on a cumulative probability. For example, PAR for probability, p, is the ratio of the instantaneous peak power, μ2, to the average power, where the probability of the instantaneous power less than μ2 is p.
According to absolute values, the PAR with probability one can be much higher. However, due to statistical distributions, the peaks, which have small occurrence probabilities, can be ignored. Thus, the clipping of the peaks with a cumulative probability less than one becomes effective on the performance. As used herein, therefore, “effective probability” means the smallest cumulative probability so that the clipping of the amplitude levels with a cumulative probability more than this cumulative probability does not affect the transmission performance considerably.
The PAR with effective probability can be more than 10 dB. Several previous attempts have been made to reduce PAR by as much as 3 dB. A summary of these attempts can be found in the book R van Nee and R. Prasad, OFDM Wireless Multimedia Communications, Artech House, 2000. The preliminary attempts focused on clipping of the multi-carrier signal. However, clipping causes in-band and out-of-band distortion which degrades the transmission performance.
Scrambling methods, such as selected mapping and partial transmit sequences, do not guarantee a PAR ratio below some low level. On the other hand, Reed-Muller codes use redundancy to reduce PAR to 3 dB along with error correction. However, Reed-Muller codes result in a considerable data rate reduction as the number of subcarriers increases. According to K. G. Paterson and V. Tarokh, “On the existence good codes with low peak-to-average power ratios,” IEEE Trans. on Information Theory, vol. 46, no. 6, September 2000, the coding distance and PAR are in contrary relation for constant amplitude modulation (such as Quadrature Phase Shift Keying and 8-Phase Shift Keying). That is, low PAR implies small coding distance, which is undesirable, and large coding distance implies high PAR, which is also undesirable.
Another method provides for considering the PAR reduction as a trellis shaping problem (see W. Henkel and B. Wagner, “Another application for trellis shaping: PAR reduction for DMT (OFDM),” IEEE Trans. on Communications, vol. 48, no. 9, September 2000). Trellis shaping adds redundancy in the transmitter so the output sequence is the output of a convolutional decoding. The redundancy is added in the transmitter through a trellis diagram so that the PAR of the output sequence becomes low. A feedback from time domain to frequency domain is employed. At the receiver, the received sequence is encoded by the corresponding convolutional encoder and then the original data sequence is recovered. However, this method selects redundant bits to reduce PAR within the structure of the used convolutional coding and hence may limit the achievable PAR reduction per the number of redundant bits. Moreover, it needs an implementation of a convolutional encoder in the receiver. Finally, a part of its complexity in the transmitter comes from the redundancy calculation according to a convolutional decoding structure.